Yoshida lifts and the Bloch–Kato conjecture for the convolution L-function

Let f1f1 (resp. f2f2) denote two (elliptic) newforms of prime level N  , trivial character and weight 2 (resp. k+2k+2, where k∈{8,12}k∈{8,12}). We provide evidence for the Bloch–Kato conjecture for the motive M=ρf1⊗ρf2(−k/2−1)M=ρf1⊗ρf2(−k/2−1) by proving that under some assumptions the ℓ-valuation of the order of the Bloch–Kato Selmer group of M is bounded from below by the ℓ-valuation of the relevant L-value (a special value of the convolution L  -function of f1f1 and f2f2). We achieve this by constructing congruences between the Yoshida lift Y(f1⊗f2)Y(f1⊗f2) of f1f1 and f2f2 and Siegel modular forms whose ℓ  -adic Galois representations are irreducible. Our result is conditional upon a conjectural formula for the Petersson norm of Y(f1⊗f2)Y(f1⊗f2).